Download Presentation
## 4. Fluid Kinematics

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**4. Fluid Kinematics**• 4.1. Velocity Field • 4.2. Continuity Equation**Fluid Kinematics**What is fluid kinematics? • Fluid kinematics is the study on fluid motion in space and time without considering the force which causes the fluid motion. • According to the continuum hypothesis the local velocity of fluid is the velocity of an infinitesimally small fluid particle/element at a given instant t. It is generally a continuous function in space and time.**How small an how large should be a fluid particle/element**in frame of the continuum concept? • The characteristic length of the fluid system of interest >> The characteristic length of a fluid particle/element >> The characteristic spacing between the molecules contained in the volume of the fluid particle/element : • For air at sea-level conditions, • - molecules in a volume of • - (λ : mean free path) • The continuum concept is valid!**4.1. Velocity Field**• Eulerian Flow Description • Lagrangian Flow Description • Streamline • Pathline • Streakline**4.1.1. In the Eulerian Method**• The flow quantities, like are described as a function of space and time without referring to any individual identity of the fluid particle :**4.1.2. Streamline**• A line in the fluid whose tangent is parallel to at a given instant t. • The family of streamlines at time t are solutions of • Where are velocity components in the respective direction**Steady flow : the streamlines are fixed in space for all**time. • Unsteady flow : the streamlines are changing from instant to instant. ▲ Fig. 4.1**4.1.3. Flow Dimensionality**• Most of the real flow are 3-dimensional and unsteady : • For many situations simplifications can be made : 2-dimensional unsteady and steady flow 1-dimensional unsteady and steady flow**4.1.4. In the Lagrangian Method**• The flow quantities are described for each individually identifiable fluid particle moving through flow field of interest. The position of the individual fluid particle is a function of time : ▲ Fig. 4.2**4.1.5. Pathline**• A line traced by an individual fluid particle : • For a steady flow the pathlines are identical with the streamlines.**4.1.6. Streakline**• A streakline consists of all fluid particles in a flow that have previously passed through a common point. Such a line can be produced by continuously injecting marked fluid (smoke in air, or dye in water) at a given location. • For steady flow : The streamline, the pathline, and the streakline are the same.**4.2. Stream-tube and Continuity Equation**• Stream-tube • Continuity Equation of a Steady Flow**4.2.1. Stream-tube**• is the surface formed instantaneously by all the streamlines that pass through a given closed curve in the fluid. ▲ Fig. 4.4**4.2.2. Continuity Equation of a Steady Flow**• For a steady flow the stream-tube formed by a closed curved fixed in space is also fixed in space, and no fluid can penetrate through the stream-tube surface, like a duct wall. ▲ Fig. 4.5**Considering a stream-tube of cylindrical cross sections**with velocities perpendicular to the cross sections and densities at the respective cross sections and assuming the velocities and densities are constant across the whole cross section , a fluid mass closed between cross section 1 and 2 at an instant t will be moved after a time interval dt by to the cross section 1’ and 2’ respectively. Because the closed mass between 1 and 2 must be the same between 1’ and 2’, and the mass between 1’ and 2 for a steady flow can not change from t and t+dt, the mass between 1 and 1’ moved in dt, must be the same as the mass between 2 and 2’ moved in the same time dt, :**Therefore the continuity equation of steady flow :**• Interpretation : The mass flow rate • through a steady stream-tube or a duct. • For incompressible fluid with : • Interpretation : The volume flow rate • From the continuity equation for incompressible fluid : • for a stream-tube. (4.1) (4.2)